Sentences of the form “No X is too Y to Z” are, in many cases, nightmares to process. The interaction of multiple negations (explicit and implicit), scalar adjectives and modals makes correctly interpreting such sentences very difficult. This has long been noted by linguists and psychologists. However, all of the accounts we can find of these types of sentences not only note the difficulty of a first-pass parse of the sentence, but also misinterpret the literal meaning.

The standard example of this kind of sentence is “No head injury is too trivial too ignore,” allegedly found on the wall of a triage department at a hospital. The usual, unreflective interpretation is as of the sentence “No head injury is trivial enough to ignore.” One hears this sentence as denying that there is some boundary along the ranking of head injuries from trivial to serious below which a head injury may responsibly be ignored. That is, no matter how trivial a head injury is, one should not ignore it.

The standardly proposed ‘literal meaning’ may be analogized to the more easily interpreted sentence “no basketball player is too short to dunk.” That is, the sentence means that no matter how trivial a head injury is, it can be ignored.

It is usually stated, then, that it follows from this literal interpretation that every head injury may be ignored. (I.e. it is claimed to follow from “No X is too Y to Z” that “Every X may be Z’d.”)

This analysis is seen in Wason and Reich’s original paper on this topic [Peter Wason and Shuli Reich, "A Verbal Illusion", The Quarterly Journal of Experimental Psychology , 1979]: “However, the injunction, Ignore all head injuries, which follows from the sentence, No head injury is too serious to be ignored, is unusual if not actually incredible,” and in a 2009 Language-log post by Mark Liberman ["No wug is too dax to be zonged", 11/28/2009]: “Against this background, we assert that T is in fact so far along the scale of smallness that there are no missiles "too small to be banned", i.e. on the not-banned side of T:

banned

not banned

—————

|T|

—————

← less small

–

more small →

It follows that all missiles should be banned.”

In conversations with Levi Roth and Bill Kowalsky, we convinced ourselves that this inference is incorrect.

There is an uninteresting way in which a counter-example to this inference can be seen: even if no head injury is too trivial to ignore, it could be that a head injury may be ignored in virtue of some other factor. For example, if an evil man with a possibility of becoming president had a head injury, it may be morally required to ignore it, for the sake of the country. What is interesting is that it does not even follow from this sentence that no head injury can be ignored simply in virtue of its location on the trivial-serious scale. To demonstrate this, we will build up this sentence from its parts, demonstrating what each contributes to the meaning of the sentence, and then provide a counter-example to the alleged inference.

Take first the complex predicate “too serious to ignore”. The extension of this predicate presupposes a scale of seriousness along which we can plot entities from most to least serious. For this predicate to have an extension, there must be some point along this spectrum, such that everything above that point cannot be ignored.

Replacing the high-end-of-the-scale predicate ‘serious’ with the low-end-of-the-scale predicate ‘trivial’, we get the complex predicate ‘too trivial to ignore’, the extension of which will be everything below a certain point on the trivial-serious scale. Everything in this extension cannot be ignored.

Now take the sentence “this head injury is too trivial to ignore.” This places a particular predicate within the previously described extension, and says of it that, in virtue of its extreme triviality, it cannot be ignored.

The sentence “No head injury is too trivial to ignore” then, states that there are no head injuries in the extension of this predicate. That is, this predicate does not provide a partition of head injuries along this scale. Every head injury falls above the boundary provided by this predicate, and so no head injury is such that it cannot be ignored in virtue of its triviality. No matter how trivial a head injury is, it may still be ignored.

However, that this predicate does not partition head injuries does not mean that no predicate does; not even that no predicate on this same scale does. That there is no boundary below which no head injury can be ignored, it does not follow that every head injury can be ignored. There may well be a boundary along this scale such that no head injury above this boundary may be ignored. Thus, there is no contradiction in the claim “No head injury is too trivial to ignore, but some head injuries are too serious to ignore.” But, the latter clause of this sentence explicitly denies the standardly perceived entailment of the former: that every head injury may be ignored.

The interest of all of this is that even with careful reflection from the people most attuned to these kinds of parsing difficulty, it has (from what I can tell) gone unnoticed for decades that the entailments of these kinds of sentences are not what have been claimed of them. An extra level of interest comes from the fact that the original, reflective analysis of the sentence (the one that entails the universal claim) appears to display the same resilience that the original, unreflective interpretation (that no head injury is trivial enough to be ignored) does. It is hard, once initiated, not to interpret the sentence as having this entailment, even when convinced that it does not. This seems to be an example of the resilience of learned intuitions.

39 Comments

Peter said,

A very nice analysis! An analogous and maybe more intuitive example: it is probably the case that “No basketball player is too tall to dunk.” But it doesn’t follow that every player can dunk; nor even that no player is prevented from dunking by their height.

(Another testament to the difficulties of complex negation: my first two attempts at the last sentence above were both wrong. I hope the current one is right!)

Michael Watts said,

I think the reason people have systematically overlooked the point made in this post is related to the difference between these example sentences:

(1) No head injury is too serious to ignore.

(2) No head injury is too trivial to ignore.

(1) is sensible, but wrong. But (2) is malformed. It's malformed because triviality is not opposed to ignoring something — ignoring a thing makes more sense as that thing becomes more trivial. But the structure under discussion can only be used for cases like (1), where ignoring a thing makes less sense as that thing becomes more serious.

Julian Hook said,

Jerry's comment implicitly makes this point, but the sentences "No head injury is too trivial to ignore" and "No basketball player is too short to dunk" do not actually have the same structure. In the first example, the NP "head injury" is the direct object of the verb "ignore"; in the second example the NP "basketball player" is the subject (not the direct object) of the verb "dunk".

Perhaps this ambiguity is part of what's confusing about these sentences: if you say "No wug is too dax to be zonged", it may not always be clear whether the wug is the supposed zonger or the zongee.

Zeppelin said,

Aaron said,

I can certainly identify with your final paragraph. For me the example sentence is reminiscent of an optical illusion that still "works" even when you know its secret. I can't stop interpreting the sentence backwards, any more than I can stop seeing an illusion designed to fool the human eye.

NSBK said,

Keeping the original syntax, what about these?
– "No head injury is too trivial to consider for treatment."
– "No head injury is too trivial to acknowledge."
It seems to me that the problem with the original literal meaning is the scale described in the post's explanation of the predicate "too trivial to ignore". Typically we ignore trivial things — the predicate only makes sense if one is looking specifically to find trivial things.

So if one considers all head injuries to be trivial, and they are searching for trivial things, then indeed "no head injury is too trivial to ignore".

Guy said,

"No head injury is too trivial to ignore" isn't "malformed" in any sense I would use the word, it's just that it seems to assume the bizarre proposition that it's conceivable that a head injury being sufficiently trivial could make it impossible (or inadvisable) to ignore it. Changing "too trivial" to "trivial enough" is probably the simplest way to "fix" it to have the apparently intended meaning while retaining the same basic structure.

This all comes across as overthinking in the extreme. We have a phrase which is widely understood with clear meaning, and a lot of professional analysis is applied to try to convince us that it is confusing.

Is it so hard to imagine that the whole is not just the sum of the parts and that certain constructions in English have their own collectively described meaning. The life of linguistics is not reason, it is experience.

D.O. said,

I am a bit surprised with this kind of analysis. Usually, grammatical constructions mean what (vast majority of) people think they mean whether it seems logical or not from someone's point of view. Anyways, trivial is a malleable concept. Even if every head injury is serious enough to be treated, it might be rather trivial for a brain surgeon. Maybe ice on the injured part and good rest is all that is required.

Lance said,

Now take the sentence “this head injury is too trivial to ignore.” This places a particular predicate within the previously described extension, and says of it that, in virtue of its extreme triviality, it cannot be ignored.

The sentence “No head injury is too trivial to ignore” then, states that there are no head injuries in the extension of this predicate.

All right. Gabriel Dupre is in some sense correct. "No student is stupid" means "There is no student S such that S is in the set of things that are stupid", or "The intersection of the set denoted by 'students' and the set denoted by 'stupid things' is the empty set". That's a pretty standard way to interpret "No". And indeed, if you interpret the sentence "no head injury is too trivial to ignore" as meaning "there is no head injury H such that H is in the set of things too trivial to ignore", or "the intersection of the set of things that are head injuries, and the set of things that are too trivial to ignore, is the empty set", then the sentence certainly does evaluate to be true.

But the reason that's true is that the set denoted by "too trivial to ignore" is itself the empty set, so naturally its intersection with the set of head injuries is empty. I feel like there's a kind of sophistry in

the extension of which will be everything below a certain point on the trivial-serious scale. Everything in this extension cannot be ignored.

in that it's accurate, but glosses over the fact that it means "there are things that are trivial, and maybe you can ignore them and maybe you can't, but the really trivial things are the ones you can't ignore". That's the part that's nonsense; that's the part that makes "too trivial to ignore" denote the empty set.

Now, granted, you could set up a context in which members of the Society for Paying Strict Attention to Really Trivial Things is discussing head injuries, and one of the members shakes her head (gently so as not to cause an injury!) and says, "No, forget about head injuries. No head injury is too trivial to ignore". In this case, yes, the sentence makes perfect sense and means what Dupre wants it to mean: there are things that are so trivial that the SPSARTT will pay attention to them, but head injuries are never those things. And that's great; but that's almost never the context of these overnegations. "No head injury is too trivial to ignore" is posted in a hospital, which is a context (read: universe, domain of quantification, whatever) in which everyone knows that the trivial things can be ignored and that no one has time to pay attention to trivial things the way the SPSARTT does. In this context–in most contexts–"No head injury is too trivial to ignore" is, as I granted above, true, but only in the same way that "No head injury is a square circle" or "No head injury is both red and not-red" is true, i.e. in a wholly uninformative way.

Does that mean that people "misinterpret the literal meaning"? Maybe so. But I'd contest that when people say "if you work through what that sentence is saying, it's wrong!" or "that sentence doesn't mean what you think it means!" or the like, they're not necessarily reacting to the literal meaning, they're reacting to the full composition of the sentence and not merely the end result. Getting a final value of "true" doesn't mean that all of the compositional semantics along the way were a good idea.

Gregory Kusnick said,

Francisco: What you're saying is that there are head injuries too trivial to be called "head injuries". Which reduces the original statement to a tautology: we call it a head injury if and only if it ought not to be ignored.

ohwilleke: Granting that the phrase means what people understand it to mean, we'd still like to have some theory about how it came to mean something apparently the opposite of what its structure suggests. That's not overthinking; seems to me that's the sort of question that an explanatory (rather than merely descriptive) science of linguistics ought to be asking.

– How much of the article would be left if we re-wrote it without all the academic speak, clearly stating the key points without all the distracting and superfluous talk about extensions of predicates and so forth?

– It may also be worth discussing in precisely what sense previous writers have missed or ignored their main point, which actually seems pretty obvious once you analyse the form through the lens of predicate logic rather than with reference to simple real-world models. I don't believe previous writers attempted to identify the logical implications of the form and failed; rather that they weren't generally looking at it as an exercise in logic in the first place. Let's not be too hard on them for ignoring technicalities that are not usually applicable in real-world circumstances.

– That said, it isn't difficult to compose sentences that do exemplify the main conclusion of the article. For example, suppose some astronomer (in science fiction, presumably) boasts: "No planet is too distant to see through my telescope." There may well be a planet — the one the astronomer is standing on — that is too close to see through the telescope, but obviously this does not contradict the astronomer's boast. What other examples can readers think of that could be used to illustrate the article's conclusion?

Viseguy said,

At first blush, this is a case where "Omit unnecessary words" would seem to apply in spades. "No head injury is trivial", or "No head injury should be ignored", or whatever. The good news, though, is that no emergency room worker in his or her right mind could possibly misinterpret the message as written. You might die while waiting to be seen for a head injury, but no ER nurse is going to read the bulletin board and bark, "Ignore that head injury — stat!" Thank heavens there's a part of the brain that can compensate for the other part of the brain that can't reliably parse "No X is too Y to Z".

That said, I can't stop thinking about Natasha Richardson. I saw her in "Cabaret" in the late '90s, and she was wonderful.

ryanwc said,

Philip said,

"Let's see if we can't come up with other examples of sentences that mean the opposite of what they say."
OK, off you go: try to come up with examples while not coming up with examples, as the original sentence literally asks you to do.

as for the post, I have a problem with this sentence:

"Now take the sentence “this head injury is too trivial to ignore.” This places a particular predicate within the previously described extension …"

But the 'previously described extension' was described as empty because no head injury was in it, so the sentence 'this head injury is too trivial to ignore' has no meaning.

J. W. Brewer said,

Adrian Morgan, I think there's a separate issue with your astronomer example. The Earth is a planet, but in ordinary discourse "planet(s)" often means in context "planet(s) other than the Earth," just as humans are a particular species of animal but "animal(s)" often means in context "non-human animal(s)." Whereas "head injury" seems unlikely to mean "head injury other than the head injury the doctor or nurse examining the patient is himself or herself currently suffering from."

Shmuel said,

ryanwc said,
The point they're really trying to make is "no head injury is too trivial to scrutinize."

Yes. And, according to Dupre's logic (although he doesn't seem to mention this point), the intended statement that "no head injury is too trivial to scrutinize" also does not literally imply that "every head injury may/should be scrutinized." After all, maybe there is some head injury that is not trivial enough to be scrutinized. (Or, to use the "uninteresting" counterexample, maybe there is some other reason to not scrutinize a particular head injury, unrelated to its measure along the scale of triviality.)

un malpaso said,

Is there a different process going on in this sentence? I parse it as, "Any fence can be jumped, no matter how high."
By this template, No head injury is too trivial to ignore" reads as, "Any head injury can be ignored, no matter how trivial," which is obviously wrong…
better to write it as, "no head injury is trivial enough to ignore," which is the underlying meaning.
In this case, "too x" is simply standing for "x of a certain measure", not the expected "x over a certain limit".

When I tried to find actual occurrences of the sentence under discussion, all that Google showed me was discussions of its meaning. But under images I found a tweet in which it is attributed to Hippocrates as cited by Ingebrigsten [sic] (1998). The name is probably meant to Ingebrigtsen, and I wonder if what we are dealing with is a putative translation from Ionic Greek into English by a Norwegian.

GH said,

What are you saying? The two sentences appear to have the same logical structure, but in one case you transform it into "CAN be ignored" and in the other into "CANNOT be jumped". What motivates this difference?

The formal analysis in the post is no doubt impeccable, but I have to admit I have a hard time following it. To me, it is simpler to try to unwind the sentence. So, consider "This head injury is too trivial to ignore." In other words, "This head injury is so trivial that we cannot ignore it" or "This head injury is so trivial that we must attend to it."

The sentence is saying that no injury like that exists. There is no head injury that must be attended to because of its extreme triviality. (But that doesn't imply anything about whether head injuries can or cannot be ignored for other reasons.)

Back to Hippocrates: the translation cited here reads: "No head injury is so trivial that it can be ignored [and none so serious that the life must be despaired of]." So it still seems to be a matter of bad translation.

@ J W Brewer: I don't agree, because the example works just as well if you imagine the astronomer and telescope are on some planet other than Earth. (It even works if the telescope is not on a planet at all but only near it, e.g. orbiting, but at the loss of some rhetorical force in my opinion.)

The purpose of the example is to identify a context where it's intuitively obvious that "no X is too Y to Z" does not entail "no X is insufficiently Y to Z" (i.e. the main point of the article). Once that point has been intuitively grasped in one context, it can be extended to others.

Actually, the first example I thought of was "No insect is too small to see through a microscope", where some insects are too large to fit on the slide, but I liked the telescope example better.

@Philip I think you might be in the running for this year's non sequita award.

Philip said,

@GH
The sentences appear to have the same logical structure, but language is not logical, according to both Wittgenstein and Humpty Dumpty. So we have pairs of seemingly opposite sentences that actually mean the same thing: I could care less/I couldn't care less; Let's see if we can't find a solution/let's see if we can find a solution.

The thinking behind the difference in CAN/CANNOT in my other post is that 'ignore' is a negative action whereas jump is a positive action. Were I to substitute 'stitch' for 'ignore', both would have Cannot after the first equals sign

I cannot in all conscience consider the sentence "This head injury is too trivial to ignore' if I am starting from the premise that 'No head injury is too trivial to ignore'.

The original sentence is ambiguous, but only if you want it to be. If most people take it to mean 'no head injuries should be ignored' then that is what it means, just like the meaning of 'I can't get no satisfaction' is always clear to people, even though some have reservations about the grammar of the sentence.

D.O. said,

There is also some flexibility in the word "injury". What rises to the level of injury and what is some physical disturbance of a lesser significance? Maybe if we (or doctors) classify something as head injury it automatically means that it should not be ignored.

Gregory Kusnick said,

Adrian: A better example might be "No chair is too big to sit in", which (unlike your astronomer's boast) has the virtue of plausibility, while at the same time leaving open the possibility of chairs too small to sit in.

Coby: Thanks for the Hippocrates quote, although your link seems to be broken. I'd be curious to know if "head injury" is an accurate translation, or if Hippocrates actually said something closer to "cranial trauma".

In the end I'm inclined to classify this as a species of nerdview. In an ER context it seems pointless to say that no head injury is too trivial to scrutinize, since every complaint that comes into an ER will receive some level of scrutiny. So the (mistranslated) Hippocrates quote seems to be functioning as a code phrase with some special technical meaning, perhaps along the lines of "Triage nurses should refer all head traumas (no matter how trivial) to an ER resident."

@Gregory: In response to your third paragraph, the ER context may involve a patient with multiple injuries, including a seemingly minor injury to the head as well as serious injuries to other parts of the body. In that context, the sign is intended as a reminder to not neglect the head injury while attending to the patient's other wounds. It isn't a superfluous reminder to not dismiss the patient altogether.

GH said,

But I don't think "ambiguity" is really relevant here. By reducing "meaning" to a single thing, you're confusing the issue. A statement can have different types of meaning: the meaning intended by the speaker/writer, what the listeners/readers understand the speaker/writer to have meant, and what they understand the statement to be "actually saying". (No doubt there are others.)

So a statement can be widely understood to mean one thing, but still be recognized as being "wrong" in the sense of technically saying something different (because it was mistakenly constructed, or perhaps used the wrong words). Language Log features examples of this regularly, particularly misnegations. Other examples include malapropisms ("a vast suppository of information"). Everyone understands what it is intended to mean, regardless of whether they realize that there's a mistake and it "really" means something different.

We let semantics override our parsing of the sentence, but can still assign some meaning to that parsing. In this case that meaning is quite definite, if challenging to arrive at. This is entirely different from cases of negative concord, idiomatic expressions or sentences where adding a negation doesn't change the meaning.

Considering this sentence, I think it's pretty obvious that no one upon seeing/hearing it actually works out all the conditions, negations and implications to figure out what the statement means, because as we have seen, it is fiendishly difficult to do so. Instead, we spot the topic words ("head injury", "trivial", "ignore"), recognize the sentence pattern "No X is too Y to Z", and fill in the meaning that seems most reasonable. Some might then have an uncomfortable feeling that it doesn't quite fit, and look closer.

I don't see why "in good conscience" you can't consider "This head injury is too trivial to ignore" as the situation the original sentence rules out. I can easily imagine a dialogue between physicians: "This head injury is too trivial to ignore, Dr. X." "I disagree, Dr. Y. No head injury is too trivial to ignore."

But then, I find your own analysis wholly unconvincing. Just because ignore might be considered a "negative action", that does not mean that "too X to ignore" should be analyzed differently from "too X to jump".

The claim simply does not hold water:

– The danger is too serious to ignore.
– The mistake is too trivial to matter.

These sentences behave just as we would expect, and "ignore" and "too trivial" follow the pattern of other verbs and adjectives. But when you put them together, you want to reverse the meaning. There's no justification for that, it's just an ad hoc fix because "too trivial to ignore" doesn't make intuitive sense otherwise. In other words, a misnegation.

Philip said,

@GH
Thanks for your considered reply. I am less interested in parsing than meaning.
What I meant was that if the starting point is 'no head injury is too trivial to ignore' then I cannot move on from that to a second statement that says 'this injury is too trivial to ignore' because the original statement tells me that no injuries fulfil that requirement, and so the second sentence cannot have any truth value and is meaningless.

I cannot easily imagine surgeon Y saying the sentence: 'This head injury is too trivial to ignore' (i.e. this head injury is so unworthy of attention that we must fail to pay attention to it) because the sentence does not make any common sense. We ignore things because they are trivial; in common sense, there could not be a situation where an increase in their triviality would lead us to stop ignoring them.

too trivial to ignore = above this threshold we can safely ignore, but the premise states that no head injuries are above this threshold, so therefore all head injuries are below the threshold and must NOT be ignored.

Some chairs ARE too high to jump though … in fact all of them are, if you want ambiguity, because no chairs can jump.